Euclidean Transport Update for the Multiplication Table

For the Euclidean decomposition of the n X n multiplication table, passage from A_n to A_{n+1} does not require recomputation of the full table. The old mass is transported exactly by re-expressing the same integers in the new base n+1, while the only genuinely new contribution comes from the newly admitted border.This note records the resulting exact update law for the joint quotient–remainder distribution, together with the corresponding transport laws for its support and parity. Multiplicity, support, and parity thereby appear as three coefficient-level readings of the same transport decomposition.